Abstract : We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, Gérard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.